Question

Which of the following will have the lowest total vapor pressure at \(25^{\circ} \mathrm{C} ?\) a. pure water (vapor pressure \(=23.8\) torr at \(25^{\circ} \mathrm{C}\) ) b. a solution of glucose in water with \(\chi_{\mathrm{C}_{\mathrm{s}} \mathrm{H}_{\mathrm{l} 2} \mathrm{O}_{\mathrm{s}}}=0.01\) c. a solution of sodium chloride in water with \(\chi_{\mathrm{NaCl}}=0.01\) d. a solution of methanol in water with \(\chi_{\mathrm{CH_{3}}, \mathrm{OH}}=0.2\) (Consider the vapor pressure of both methanol \([143\) torr at \(\left.25^{\circ} \mathrm{C}\right]\) and water.

Short answer

The lowest total vapor pressure at \(25^{\circ} \mathrm{C}\) is for solutions b and c (solution of glucose in water with \(\chi_{C_{6}H_{12}O_{6}} = 0.01\) and solution of sodium chloride in water with \(\chi_{NaCl} = 0.01\)), with a vapor pressure of 23.562 torr.

Step-by-step solution

Expression for Raoult's Law

To calculate the vapor pressure of a solution, we use Raoult's Law: \(P_{solution} = P_{solvent} * \chi_{solvent}\), where \(P_{solution}\) is the vapor pressure of the solution, \(P_{solvent}\) is the vapor pressure of the pure solvent, and \(\chi_{solvent}\) is the mole fraction of the solvent in the solution.

Calculate the mole fraction of solvent for each solution

We are given the mole fractions for the solutes in each solution. To find the mole fraction of the solvent, we can subtract the mole fraction of the solute from 1: a. \(\chi_{water} = 1\) b. \(\chi_{water} = 1 - \chi_{C_{6}H_{12}O_{6}} = 1 - 0.01 = 0.99\) c. \(\chi_{water} = 1 - \chi_{NaCl} = 1 - 0.01 = 0.99\) d. \(\chi_{water} = 1 - \chi_{CH_{3}OH} = 1 - 0.2 = 0.8\)

Calculate the vapor pressure for each solution using Raoult's Law

Now we can plug our mole fractions and vapor pressures into Raoult's Law to find the vapor pressure of each solution: a. \(P_{water} = 23.8 * 1 = 23.8\) torr b. \(P_{glucose} = 23.8 * 0.99 = 23.562\) torr c. \(P_{NaCl} = 23.8 * 0.99 = 23.562\) torr d. For the methanol solution, we need to consider both the vapor pressures of methanol and water. We apply Raoult's Law separately for each component and then add the results: \(P_{CH_{3}OH} = 143 * 0.2 = 28.6\) torr and \(P_{water} = 23.8 * 0.8 = 19.04\) torr. So, \(P_{methanol} = P_{CH_{3}OH} + P_{water} = 28.6 + 19.04 = 47.64\) torr.

Compare the vapor pressures and determine the lowest

Comparing the vapor pressures, we observe that the glucose solution (b) and the sodium chloride solution (c) have the lowest vapor pressures at 23.562 torr. Hence, the solutions b and c (solution of glucose in water with \(\chi_{C_{6}H_{12}O_{6}} = 0.01\) and solution of sodium chloride in water with \(\chi_{NaCl} = 0.01\)) have the lowest total vapor pressure.

Key concepts

Vapor Pressure

Vapor pressure is a critical concept in chemistry, especially when studying the behavior of liquids and solutions. In simple terms, it's the pressure exerted by a vapor in equilibrium with its liquid phase at a given temperature. The higher the vapor pressure of a liquid, the more volatile it is, meaning it will evaporate or boil more easily.

At a constant temperature, pure liquids have a specific vapor pressure; however, when a solute is dissolved into a liquid to form a solution, the vapor pressure changes. This change is predicted by Raoult's Law, which states that the vapor pressure of a solvent above a solution is directly proportional to its mole fraction in the solution. Applying Raoult's Law helps in understanding how solutes can lower the vapor pressure of a solvent through the phenomenon called vapor pressure lowering, a type of colligative property.

Mole Fraction

Mole fraction is a way of expressing the concentration of a component in a mixture or solution. It's defined as the ratio of the number of moles of a particular component to the total number of moles of all components in the mixture.

For a solution with a solute A and solvent B, the mole fraction of A, denoted as \(\chi_A\), is calculated using the formula \(\chi_A = \frac{n_A}{n_A + n_B}\), where \(n_A\) and \(n_B\) represent the number of moles of the solute and solvent, respectively. The mole fraction is dimensionless and always less than or equal to one. It's particularly useful in Raoult's Law because it reflects the ratio of solvent to solution, thus determining the change in vapor pressure upon the addition of a solute.

Solutions Chemistry

Solutions chemistry revolves around the study of homogeneous mixtures composed of two or more substances. In a solution, the substance present in the largest amount is called the solvent, while the other substances are solutes. When solutes dissolve in a solvent, they can alter the physical properties of the solvent, such as vapor pressure, boiling point, and freezing point.

Solutions can be solid, liquid, or gas, but in the context of Raoult's Law, we're typically referring to liquid solutions. Understanding the quantitative aspects of solutions, including how solutes affect vapor pressure and mole fractions, enables chemists and students to predict how different compositions will behave under various conditions.

Colligative Properties

Colligative properties are characteristics of solutions that depend only on the ratio of the number of solute particles to the number of solvent molecules in a solution, not on the nature of the chemical species present. These properties include vapor pressure lowering, boiling point elevation, freezing point depression, and osmotic pressure.

To delve deeper into one of these properties: vapor pressure lowering is illustrated by Raoult's Law, which shows that the vapor pressure of a solvent will decrease when a non-volatile solute is added. This decrease is directly proportional to the mole fraction of the solvent in the solution. These concepts are pivotal for many practical applications, including the formulation of antifreeze solutions, the manufacturing of pharmaceuticals, and the design of chemical separation processes.